# Octagon > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Octagon > Markdown URL: https://mediated.wiki/source/Octagon.md > Source: https://en.wikipedia.org/wiki/Octagon > Source revision: 1357145679 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) Polygon shape with eight sides "Octagonal" redirects here. For other uses, see [Octagon (disambiguation)](/source/Octagon_(disambiguation)) and [Octagonal (disambiguation)](/source/Octagonal_(disambiguation)). Regular octagon A regular octagon Type Regular polygon Edges and vertices 8 Schläfli symbol {8}, t{4} Coxeter–Dynkin diagrams Symmetry group Dihedral (D8), order 2×8 Internal angle (degrees) 135° Properties Convex, cyclic, equilateral, isogonal, isotoxal Dual polygon Self In [geometry](/source/Geometry), an **octagon** (from [Ancient Greek](/source/Ancient_Greek_language) ὀκτάγωνον*(*oktágōnon*)* 'eight angles') is an eight-sided [polygon](/source/Polygon) or 8-gon. A *[regular](/source/Regular_polygon) octagon* has [Schläfli symbol](/source/Schl%C3%A4fli_symbol) {8}[1] and can also be constructed as a quasiregular [truncated](/source/Truncation_(geometry)) [square](/source/Square), t{4}, which alternates two types of edges. A truncated octagon, t{8} is a [hexadecagon](/source/Hexadecagon), {16}. A 3D analog of the octagon can be the [rhombicuboctahedron](/source/Rhombicuboctahedron) with the triangular faces on it like the replaced edges, if one considers the octagon to be a truncated square. ## Properties The diagonals of the green [quadrilateral](/source/Quadrilateral) are equal in length and at right angles to each other Assuming [plane geometry](/source/Plane_geometry), the sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a quadrilateral that is both [equidiagonal](/source/Equidiagonal_quadrilateral) and [orthodiagonal](/source/Orthodiagonal_quadrilateral) (that is, whose diagonals are equal in length and at right angles to each other).[2]: Prop. 9 The [midpoint octagon](/source/Midpoint_polygon) of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. If squares are constructed all internally or all externally on the sides of the midpoint octagon, then the midpoints of the segments connecting the centers of opposite squares themselves form the vertices of a square.[2]: Prop. 10 ### Regularity A [regular](/source/Regular_polygon) octagon is a closed [figure](/source/Shape) with sides of the same length and internal angles of the same size. It has eight lines of [reflective symmetry](/source/Reflective_symmetry) and [rotational symmetry](/source/Rotational_symmetry) of order 8. A regular octagon is represented by the [Schläfli symbol](/source/Schl%C3%A4fli_symbol) {8}. The internal [angle](/source/Angle) at each vertex of a regular octagon is 135[°](/source/Degree_(angle)) ( 3 π 4 {\displaystyle \scriptstyle {\frac {3\pi }{4}}} [radians](/source/Radian)). The [central angle](/source/Central_angle) is 45° ( π 4 {\displaystyle \scriptstyle {\frac {\pi }{4}}} radians). ### Area The area of a regular octagon of side length *a* is given by - A = 2 cot ⁡ π 8 a 2 = 2 ( 1 + 2 ) a 2 ≈ 4.828 a 2 . {\displaystyle A=2\cot {\frac {\pi }{8}}a^{2}=2(1+{\sqrt {2}})a^{2}\approx 4.828\,a^{2}.} In terms of the [circumradius](/source/Circumscribed_circle) *R*, the area is - A = 4 sin ⁡ π 4 R 2 = 2 2 R 2 ≈ 2.828 R 2 . {\displaystyle A=4\sin {\frac {\pi }{4}}R^{2}=2{\sqrt {2}}R^{2}\approx 2.828\,R^{2}.} In terms of the [apothem](/source/Apothem) *r* (see also [inscribed figure](/source/Inscribed_figure)), the area is - A = 8 tan ⁡ π 8 r 2 = 8 ( 2 − 1 ) r 2 ≈ 3.314 r 2 . {\displaystyle A=8\tan {\frac {\pi }{8}}r^{2}=8({\sqrt {2}}-1)r^{2}\approx 3.314\,r^{2}.} These last two [coefficients](/source/Coefficients) bracket the value of [pi](/source/Pi), the area of the [unit circle](/source/Unit_circle). The [area](/source/Area) of a [regular](/source/Regular_polygon) octagon can be computed as a [truncated](/source/Truncation_(geometry)) [square](/source/Square_(geometry)). The area can also be expressed as - A = S 2 − a 2 , {\displaystyle \,\!A=S^{2}-a^{2},} where *S* is the span of the octagon, or the second-shortest diagonal; and *a* is the length of one of the sides, or bases. This is easily proven if one takes an octagon, draws a square around the outside (making sure that four of the eight sides overlap with the four sides of the square) and then takes the corner triangles (these are [45–45–90 triangles](/source/Special_right_triangles#45–45–90_triangle)) and places them with right angles pointed inward, forming a square. The edges of this square are each the length of the base. Given the length of a side *a*, the span *S* is - S = a 2 + a + a 2 = ( 1 + 2 ) a ≈ 2.414 a . {\displaystyle S={\frac {a}{\sqrt {2}}}+a+{\frac {a}{\sqrt {2}}}=(1+{\sqrt {2}})a\approx 2.414a.} The span, then, is equal to the *[silver ratio](/source/Silver_ratio)* times the side, a. The area is then as above: - A = ( ( 1 + 2 ) a ) 2 − a 2 = 2 ( 1 + 2 ) a 2 ≈ 4.828 a 2 . {\displaystyle A=((1+{\sqrt {2}})a)^{2}-a^{2}=2(1+{\sqrt {2}})a^{2}\approx 4.828a^{2}.} Expressed in terms of the span, the area is - A = 2 ( 2 − 1 ) S 2 ≈ 0.828 S 2 . {\displaystyle A=2({\sqrt {2}}-1)S^{2}\approx 0.828S^{2}.} Another simple formula for the area is - A = 2 a S . {\displaystyle \ A=2aS.} More often the span *S* is known, and the length of the sides, *a*, is to be determined, as when cutting a square piece of material into a regular octagon. From the above, - a ≈ S / 2.414. {\displaystyle a\approx S/2.414.} The two end lengths *e* on each side (the leg lengths of the triangles (green in the image) truncated from the square), as well as being e = a / 2 , {\displaystyle e=a/{\sqrt {2}},} may be calculated as - e = ( S − a ) / 2. {\displaystyle \,\!e=(S-a)/2.} ### Circumradius and inradius The [circumradius](/source/Circumradius) of the regular octagon in terms of the side length *a* is[3] - R = ( 4 + 2 2 2 ) a ≈ 1.307 a , {\displaystyle R=\left({\frac {\sqrt {4+2{\sqrt {2}}}}{2}}\right)a\approx 1.307a,} and the [inradius](/source/Inradius) is - r = ( 1 + 2 2 ) a ≈ 1.207 a . {\displaystyle r=\left({\frac {1+{\sqrt {2}}}{2}}\right)a\approx 1.207a.} (that is one-half the *[silver ratio](/source/Silver_ratio)* times the side, *a*, or one-half the span, *S*) The inradius can be calculated from the circumradius as - r = R cos ⁡ π 8 {\displaystyle r=R\cos {\frac {\pi }{8}}} ### Diagonality The regular octagon, in terms of the side length *a*, has three different types of [diagonals](/source/Diagonal): - Short diagonal; - Medium diagonal (also called span or height), which is twice the length of the inradius; - Long diagonal, which is twice the length of the circumradius. The formula for each of them follows from the basic principles of geometry. Here are the formulas for their length:[4] - Short diagonal: a 2 + 2 {\displaystyle a{\sqrt {2+{\sqrt {2}}}}} ; - Medium diagonal: ( 1 + 2 ) a {\displaystyle (1+{\sqrt {2}})a} ; (*[silver ratio](/source/Silver_ratio)* times a) - Long diagonal: a 4 + 2 2 {\displaystyle a{\sqrt {4+2{\sqrt {2}}}}} . ### Construction building a regular octagon by folding a sheet of paper A regular octagon at a given circumcircle may be constructed as follows: 1. Draw a circle and a diameter AOE, where O is the center and A, E are points on the circumcircle. 1. Draw another diameter GOC, perpendicular to AOE. 1. (Note in passing that A,C,E,G are vertices of a square). 1. Draw the bisectors of the right angles GOA and EOG, making two more diameters HOD and FOB. 1. A,B,C,D,E,F,G,H are the vertices of the octagon. Octagon at a given circumcircle Octagon at a given side length, animation (The construction is very similar to that of [hexadecagon at a given side length](/source/Hexadecagon#Construction).) A regular octagon can be constructed using a [straightedge](/source/Straightedge) and a [compass](/source/Compass_(drawing_tool)), as 8 = 23, a [power of two](/source/Power_of_two): Meccano octagon construction. The regular octagon can be constructed with [meccano](/source/Meccano) bars. Twelve bars of size 4, three bars of size 5 and two bars of size 6 are required. Each side of a regular octagon subtends half a right angle at the centre of the circle which connects its vertices. Its area can thus be computed as the sum of eight isosceles triangles, leading to the result: - Area = 2 a 2 ( 2 + 1 ) {\displaystyle {\text{Area}}=2a^{2}({\sqrt {2}}+1)} for an octagon of side *a*. ### Standard coordinates The coordinates for the vertices of a regular octagon centered at the origin and with side length 2 are: - (±1, ±(1+√2)) - (±(1+√2), ±1). ### Dissectibility 8-cube projection 24 rhomb dissection Regular Isotoxal [Coxeter](/source/Coxeter) states that every [zonogon](/source/Zonogon) (a 2*m*-gon whose opposite sides are parallel and of equal length) can be dissected into *m*(*m*-1)/2 parallelograms.[5] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the *regular octagon*, *m*=4, and it can be divided into 6 rhombs, with one example shown below. This decomposition can be seen as 6 of 24 faces in a [Petrie polygon](/source/Petrie_polygon) projection plane of the [tesseract](/source/Tesseract). The list (sequence [A006245](https://oeis.org/A006245) in the [OEIS](/source/On-Line_Encyclopedia_of_Integer_Sequences)) defines the number of solutions as eight, by the eight orientations of this one dissection. These squares and rhombs are used in the [Ammann–Beenker tilings](/source/Ammann%E2%80%93Beenker_tiling). Regular octagon dissected Tesseract 4 rhombs and 2 squares ## Skew A regular skew octagon seen as edges of a [square antiprism](/source/Square_antiprism), symmetry D4d, [2+,8], (2*4), order 16. A **skew octagon** is a [skew polygon](/source/Skew_polygon) with eight vertices and edges but not existing on the same plane. The interior of such an octagon is not generally defined. A *skew zig-zag octagon* has vertices alternating between two parallel planes. A **regular skew octagon** is [vertex-transitive](/source/Vertex-transitive) with equal edge lengths. In three dimensions it is a zig-zag skew octagon and can be seen in the vertices and side edges of a [square antiprism](/source/Square_antiprism) with the same D4d, [2+,8] symmetry, order 16. ### Petrie polygons The regular skew octagon is the [Petrie polygon](/source/Petrie_polygon) for these higher-dimensional regular and [uniform polytopes](/source/Uniform_polytope), shown in these skew [orthogonal projections](/source/Orthogonal_projection) of in A7, B4, and D5 [Coxeter planes](/source/Coxeter_plane). A7 D5 B4 7-simplex 5-demicube 16-cell Tesseract ## Symmetry Symmetry The eleven symmetries of a regular octagon. Lines of reflections are blue through vertices, purple through edges, and gyration orders are given in the center. Vertices are colored by their symmetry position. The *regular octagon* has Dih8 symmetry, order 16. There are three dihedral subgroups: Dih4, Dih2, and Dih1, and four [cyclic subgroups](/source/Cyclic_group): Z8, Z4, Z2, and Z1, the last implying no symmetry. Example octagons by symmetry r16 d8 g8 p8 d4 g4 p4 d2 g2 p2 a1 On the regular octagon, there are eleven distinct symmetries. John Conway labels full symmetry as **r16**.[6] The dihedral symmetries are divided depending on whether they pass through vertices (**d** for diagonal) or edges (**p** for perpendiculars) Cyclic symmetries in the middle column are labeled as **g** for their central gyration orders. Full symmetry of the regular form is **r16** and no symmetry is labeled **a1**. The most common high symmetry octagons are **p8**, an [isogonal](/source/Isogonal_figure) octagon constructed by four mirrors can alternate long and short edges, and **d8**, an [isotoxal](/source/Isotoxal_figure) octagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are [duals](/source/Dual_polygon) of each other and have half the symmetry order of the regular octagon. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the **g8** subgroup has no degrees of freedom but can be seen as [directed edges](/source/Directed_edge). ## Use The octagonal floor plan, [Dome of the Rock](/source/Dome_of_the_Rock), [Jerusalem](/source/Jerusalem). The octagonal shape is used as a design element in architecture. The [Dome of the Rock](/source/Dome_of_the_Rock) has a characteristic octagonal plan. The [Tower of the Winds](/source/Tower_of_the_Winds) in Athens is another example of an octagonal structure. The octagonal plan has also been in church architecture such as [St. George's Cathedral, Addis Ababa](/source/St._George's_Cathedral%2C_Addis_Ababa), [Basilica of San Vitale](/source/Basilica_of_San_Vitale) (in Ravenna, Italia), [Castel del Monte](/source/Castel_del_Monte%2C_Apulia) (Apulia, Italia), [Florence Baptistery](/source/Florence_Baptistery), [Zum Friedefürsten Church](/source/Zum_Friedef%C3%BCrsten_Church) (Germany) and a number of [octagonal churches in Norway](/source/Octagonal_churches_in_Norway). The central space in the [Aachen Cathedral](/source/Aachen_Cathedral), the Carolingian [Palatine Chapel](/source/Palatine_Chapel%2C_Aachen), has a regular octagonal floorplan. Uses of octagons in churches also include lesser design elements, such as the octagonal [apse](/source/Apse) of [Nidaros Cathedral](/source/Nidaros_Cathedral). Architects such as [John Andrews](/source/John_Andrews_(architect)) have used octagonal floor layouts in buildings for functionally separating office areas from building services, such as in the [Intelsat Headquarters](/source/Intelsat_Headquarters) of Washington or [Callam Offices](https://en.wikipedia.org/w/index.php?title=Callam_Offices&action=edit&redlink=1) in Canberra. - [Umbrellas](/source/Umbrella) often have an octagonal outline. - The famous [Bukhara rug](/source/Bukhara_rug) design incorporates an octagonal "elephant's foot" motif. - The street & block layout of [Barcelona](/source/Barcelona)'s [Eixample](/source/Eixample) district is based on non-regular octagons - [Janggi](/source/Janggi) uses octagonal pieces. - Japanese [lottery machines](/source/Lottery_machine) often have octagonal shape. - A [Stop sign](/source/Stop_sign) used in [English](/source/English_language)-speaking countries, as well as in most [European countries](/source/European_countries) - The trigrams of the [Taoist](/source/Taoism) *[bagua](/source/Bagua)* are often arranged octagonally - Famous octagonal gold cup from the [Belitung shipwreck](/source/Belitung_shipwreck) - Classes at [Shimer College](/source/Shimer_College) are traditionally held around octagonal tables - The [Labyrinth of the Reims Cathedral](/source/Labyrinth_of_the_Reims_Cathedral) with a quasi-octagonal shape. - The movement of the [analog stick](/source/Analog_stick)(s) of the [Nintendo 64 controller](/source/Nintendo_64_controller), the [GameCube controller](/source/GameCube_controller), the [Wii Nunchuk](/source/Wii_Nunchuk) and the [Classic Controller](/source/Classic_Controller) is bounded by an octagonal frame, helping the user aim the stick in [cardinal directions](/source/Cardinal_direction) while still allowing circular freedom. - Chair from [A la Ronde](/source/A_la_Ronde), with octagonal seats and backs (set of eight) ## Derived figures - The [truncated square tiling](/source/Truncated_square_tiling) has 2 octagons around every vertex. - An [octagonal prism](/source/Octagonal_prism) contains two octagonal faces. - An [octagonal antiprism](/source/Octagonal_antiprism) contains two octagonal faces. - The [truncated cuboctahedron](/source/Truncated_cuboctahedron) contains 6 octagonal faces. - The [omnitruncated cubic honeycomb](/source/Omnitruncated_cubic_honeycomb) ### Related polytopes The *octagon*, as a [truncated](/source/Truncation_(geometry)) [square](/source/Square), is first in a sequence of truncated [hypercubes](/source/Hypercube): Truncated hypercubes Image ... Name Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube Coxeter diagram Vertex figure ( )v( ) ( )v{ } ( )v{3} ( )v{3,3} ( )v{3,3,3} ( )v{3,3,3,3} ( )v{3,3,3,3,3} As an [expanded](/source/Expansion_(geometry)) square, it is also first in a sequence of expanded hypercubes: Expanded hypercubes ... Octagon Rhombicuboctahedron Runcinated tesseract Stericated 5-cube Pentellated 6-cube Hexicated 7-cube Heptellated 8-cube ## See also - [Bumper pool](/source/Bumper_pool) - [Hansen's small octagon](/source/Hansen's_small_octagon) - [Octagon house](/source/Octagon_house) - [Octagonal number](/source/Octagonal_number) - [Octagram](/source/Octagram) - [Octahedron](/source/Octahedron), 3D shape with eight faces. - [Oktogon](/source/Oktogon_(intersection)), a major intersection in [Budapest](/source/Budapest), [Hungary](/source/Hungary) - [Rub el Hizb](/source/Rub_el_Hizb) (also known as Al Quds Star and as Octa Star), a common motif in [Islamic architecture](/source/Islamic_architecture) - [Smoothed octagon](/source/Smoothed_octagon) ## References 1. **[^](#cite_ref-1)** Wenninger, Magnus J. (1974), [*Polyhedron Models*](https://books.google.com/books?id=N8lX2T-4njIC&pg=PA9), Cambridge University Press, p. 9, [ISBN](/source/ISBN_(identifier)) [9780521098595](https://en.wikipedia.org/wiki/Special:BookSources/9780521098595). 1. ^ [***a***](#cite_ref-Oai_2-0) [***b***](#cite_ref-Oai_2-1) Dao Thanh Oai (2015), "Equilateral triangles and Kiepert perspectors in complex numbers", *Forum Geometricorum* 15, 105--114. [http://forumgeom.fau.edu/FG2015volume15/FG201509index.html](http://forumgeom.fau.edu/FG2015volume15/FG201509index.html) [Archived](https://web.archive.org/web/20150705033424/http://forumgeom.fau.edu/FG2015volume15/FG201509index.html) 2015-07-05 at the [Wayback Machine](/source/Wayback_Machine) 1. **[^](#cite_ref-3)** Weisstein, Eric. "Octagon." From MathWorld--A Wolfram Web Resource. [http://mathworld.wolfram.com/Octagon.html](http://mathworld.wolfram.com/Octagon.html) 1. **[^](#cite_ref-4)** Alsina, Claudi; Nelsen, Roger B. (2023), [*A Panoply of Polygons*](https://books.google.com/books?id=LqatEAAAQBAJ&pg=PA124), Dolciani Mathematical Expositions, vol. 58, American Mathematical Society, p. 124, [ISBN](/source/ISBN_(identifier)) [9781470471842](https://en.wikipedia.org/wiki/Special:BookSources/9781470471842) 1. **[^](#cite_ref-5)** [Coxeter](/source/Coxeter), Mathematical recreations and Essays, Thirteenth edition, p.141 1. **[^](#cite_ref-6)** John H. Conway, Heidi Burgiel, [Chaim Goodman-Strauss](/source/Chaim_Goodman-Strauss), (2008) The Symmetries of Things, [ISBN](/source/ISBN_(identifier)) [978-1-56881-220-5](https://en.wikipedia.org/wiki/Special:BookSources/978-1-56881-220-5) (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278) ## External links Look up ***[octagon](https://en.wiktionary.org/wiki/Special:Search/octagon)*** in Wiktionary, the free dictionary. - [Octagon Calculator](http://rechneronline.de/pi/octagon.php) - [Definition and properties of an octagon](http://www.mathopenref.com/octagon.html) With interactive animation v t e Polygons (List) Triangles Acute Equilateral Ideal Isosceles Kepler Obtuse Right Quadrilaterals Antiparallelogram Apollonius Bicentric Crossed Cyclic Equidiagonal Ex-tangential Harmonic Isosceles trapezoid Kite Orthodiagonal Parallelogram Rectangle Right kite Right trapezoid Rhomboid Rhombus Square Tangential Tangential trapezoid Trapezoid By number of sides 1–10 sides Monogon (1) Digon (2) Triangle (3) Quadrilateral (4) Pentagon (5) Hexagon (6) Heptagon/Septagon (7) Octagon (8) Nonagon/Enneagon (9) Decagon (10) 11–20 sides Hendecagon (11) Dodecagon (12) Tridecagon (13) Tetradecagon (14) Pentadecagon (15) Hexadecagon (16) Heptadecagon (17) Octadecagon (18) Icosagon (20) >20 sides Icositrigon (23) Icositetragon (24) Triacontagon (30) 257-gon Chiliagon (1000) Myriagon (10,000) 65537-gon Megagon (1,000,000) Apeirogon (∞) Star polygons Pentagram Hexagram Heptagram Octagram Enneagram Decagram Hendecagram Dodecagram Classes Concave Convex Cyclic Equiangular Equilateral Infinite skew Isogonal Isotoxal Magic Pseudotriangle Rectilinear Regular Reinhardt Simple Skew Star-shaped Tangential Weakly simple --- Adapted from the Wikipedia article [Octagon](https://en.wikipedia.org/wiki/Octagon) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Octagon?action=history)). 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